You are talking about qualitative analysis, where different chemical tests are performed to test for the presence of particular ions present in an unknown solution. The resulting colors or precipitates will determine if the test is positive or negative. We can use this to test for either cations or anions. Since you did list any specific test results, I will talk about tests for anions in general.
The easiest anions to test for are halides. Simply add silver nitrate to the solution. Silver halides precipitate from solution and the color will tell you which halide it is (AgCl is white, AgBr is light yellow, and AgI is darker yellow).
Carbonate and bicarbonate are also easy to test for. Simply add some strong acid like HCl. If these anions are present, the acid will cause them to form CO2 which will very obviously bubble from the solution.
Acetate can be tested for also by the addition of strong acid. It will cause acetic acid to form, which has a characteristic vinegar smell.
Nitrites and nitrates can also be tested with the addition of strong acid. They will produce NO2 gas which has a reddish brown color.
Also, the sulfate ion can be tested by the addition of a barium salt. This will form BaSO4, which is completely insoluble in any aqueous medium.
The limitations of these tests are that some anions give the same result, like nitrite and nitrate, and carbonate and bicarbonate as mentioned above. Additional, more complicated follow up tests have to be done to distinguish between them.